Quadrupedal animals show remarkable capabilities in traversing diverse terrains and display a range of behaviors and gait patterns. Achieving similar performance is a key goal for robotics researchers. We propose a bio-inspired approach to the design of quadrupeds that seeks to exploit the body and the passive properties of the robot. Using this novel approach we develop \textit{PAWS}, a Passive Automata With Synergies. By leveraging the principles of motor synergies, the design incorporates variable stiffness, biological anatomical insights, and self-organization to simplify control while maximizing its capabilities. The resulting synergy-based quadruped requires only four actuators and exhibits emergent, animal-like dynamical responses, including robustness to environmental perturbations and a wide range of behaviors. The finding contributes to the development of machine intelligence, and provides robots with more efficient and natural-looking robotic locomotion by combining synergistic actuation, compliant body properties, and embodied compensatory strategies.